Wicksell's Problem in Local Stereology

• Published in: Advances in applied probability Vol. 45; no. 4; pp. 925 - 944
• Main Authors: Thórisdóttir, Ó., Kiderlen, M.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01-12-2013; Applied Probability Trust
• Subjects: 45Q05; 60D05; 62G05; 65R30; Algorithms; Approximation; Attraction; Computer simulation; Cumulative distribution functions; Differential equations; Distribution functions; Geometric planes; Hyperplanes; Integral equations; inverse problem; local stereology; Mathematical constants; Mathematical models; Mathematical problems; numerical unfolding; Particle size distribution; Probability; Probability distribution; Random variables; Size distribution; Stereology; stereology of extremes; Stochastic Geometry and Statistical Applications; Studies; Uniqueness; Wicksell's corpuscle problem; 60D05; 62G05; inverse problem; numerical unfolding; 65R30; 45Q05; local stereology; stereology of extremes; Wicksell's corpuscle problem
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1239/aap/1386857851

Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.